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Mirrors > Home > MPE Home > Th. List > eldprdi | Structured version Visualization version GIF version |
Description: The domain of definition of the internal direct product, which states that 𝑆 is a family of subgroups that mutually commute and have trivial intersections. (Contributed by Mario Carneiro, 25-Apr-2016.) (Revised by AV, 14-Jul-2019.) |
Ref | Expression |
---|---|
eldprdi.0 | ⊢ 0 = (0g‘𝐺) |
eldprdi.w | ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } |
eldprdi.1 | ⊢ (𝜑 → 𝐺dom DProd 𝑆) |
eldprdi.2 | ⊢ (𝜑 → dom 𝑆 = 𝐼) |
eldprdi.3 | ⊢ (𝜑 → 𝐹 ∈ 𝑊) |
Ref | Expression |
---|---|
eldprdi | ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldprdi.1 | . 2 ⊢ (𝜑 → 𝐺dom DProd 𝑆) | |
2 | eldprdi.3 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑊) | |
3 | eqid 2733 | . . 3 ⊢ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹) | |
4 | oveq2 7417 | . . . 4 ⊢ (𝑓 = 𝐹 → (𝐺 Σg 𝑓) = (𝐺 Σg 𝐹)) | |
5 | 4 | rspceeqv 3634 | . . 3 ⊢ ((𝐹 ∈ 𝑊 ∧ (𝐺 Σg 𝐹) = (𝐺 Σg 𝐹)) → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) |
6 | 2, 3, 5 | sylancl 587 | . 2 ⊢ (𝜑 → ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)) |
7 | eldprdi.2 | . . 3 ⊢ (𝜑 → dom 𝑆 = 𝐼) | |
8 | eldprdi.0 | . . . 4 ⊢ 0 = (0g‘𝐺) | |
9 | eldprdi.w | . . . 4 ⊢ 𝑊 = {ℎ ∈ X𝑖 ∈ 𝐼 (𝑆‘𝑖) ∣ ℎ finSupp 0 } | |
10 | 8, 9 | eldprd 19874 | . . 3 ⊢ (dom 𝑆 = 𝐼 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))) |
11 | 7, 10 | syl 17 | . 2 ⊢ (𝜑 → ((𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆) ↔ (𝐺dom DProd 𝑆 ∧ ∃𝑓 ∈ 𝑊 (𝐺 Σg 𝐹) = (𝐺 Σg 𝑓)))) |
12 | 1, 6, 11 | mpbir2and 712 | 1 ⊢ (𝜑 → (𝐺 Σg 𝐹) ∈ (𝐺 DProd 𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∃wrex 3071 {crab 3433 class class class wbr 5149 dom cdm 5677 ‘cfv 6544 (class class class)co 7409 Xcixp 8891 finSupp cfsupp 9361 0gc0g 17385 Σg cgsu 17386 DProd cdprd 19863 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-ixp 8892 df-dprd 19865 |
This theorem is referenced by: dprdfsub 19891 dprdf11 19893 dprdsubg 19894 dprdub 19895 dpjidcl 19928 |
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